Flexible intuitions of Euclidean geometry in anAmazonian indigene groupVéronique Izarda,b,c,1,2, Pierre Picad,e,1, Elizabeth S. Spelkec, and Stanislas Dehaenef,g,h,i

aLaboratoire Psychologie de la Perception, Université Paris Descartes, 75006 Paris, France; bCNRS UMR 8158, 75006 Paris, France; cDepartment of Psychology,Harvard University, Cambridge, MA 02138; dLaboratoire Structure Formelle du Langage, Université Paris 8, 93200 Saint-Denis, France; eCNRS UMR 7023,93200 Saint Denis, France; fCognitive Neuroimaging Unit, Institut National de la Santé et de la Recherche Médicale, F-91191 Gif-sur-Yvette, France;gCommissariat à l’Energie Atomique, I2BM, NeuroSpin, F-91191 Gif-sur-Yvette, France; hUniversité Paris-Sud, F-91405 Orsay, France; and iCollège deFrance, 75005 Paris, France

Edited* by Charles R. Gallistel, Rutgers University, Piscataway, NJ, and approved April 25, 2011 (received for review November 12, 2010)

Kant argued that Euclidean geometry is synthesized on the basisof an a priori intuition of space. This proposal inspired muchbehavioral research probing whether spatial navigation in humansand animals conforms to the predictions of Euclidean geometry.However, Euclidean geometry also includes concepts that tran-scend the perceptible, such as objects that are infinitely small orinfinitely large, or statements of necessity and impossibility. Wetested the hypothesis that certain aspects of nonperceptibleEuclidian geometry map onto intuitions of space that are presentin all humans, even in the absence of formal mathematicaleducation. Our tests probed intuitions of points, lines, and surfacesin participants from an indigene group in the Amazon, theMundurucu, as well as adults and age-matched children controlsfrom the United States and France and younger US childrenwithout education in geometry. The responses of Mundurucuadults and children converged with that of mathematically edu-cated adults and children and revealed an intuitive understandingof essential properties of Euclidean geometry. For instance, on asurface described to them as perfectly planar, the Mundurucu’sestimations of the internal angles of triangles added up to ∼180degrees, and when asked explicitly, they stated that there existsone single parallel line to any given line through a given point.These intuitionswere also partially in place in the group of youngerUS participants. We conclude that, during childhood, humans de-velop geometrical intuitions that spontaneously accord with theprinciples of Euclidean geometry, even in the absence of trainingin mathematics.

mathematical cognition | spatial cognition | culture

Geometry . . . does not deal with natural solids, but with ideal, ab-solutely invariable ones. These ideal bodies are complete fabricationsof our mind, and experience only provides an occasion that incites usto draw it out.

Henri Poincaré, L’espace et la géométrie (1)

Although Kant’s argument for the existence of an a priori in-tuition of space (2, 3) is philosophical in nature, it implies

that the human mind is spontaneously endowed with Euclideanintuitions, an empirically testable proposal that belongs to cog-nitive science. Indeed, research has shown that human adults,children, and animals are sensitive to the geometric properties ofspace and use these properties to recognize objects and forms (4–6) or navigate through the environment (7–11). These achieve-ments are supported by an increasingly well-understood neuralcircuitry, including head direction and grid cells that provide anapproximately planar coordinate system for space (12, 13).Do these spatial representations endow humans and other

animal species with an inherently Euclidean mental geometry assuggested by Kant? Although many species spontaneously adoptchoices that are optimal in Euclidean geometry, such as follow-ing the shortest straight path to return to a given location (14–19), animal and human perception of space has been found to

violate Euclidean principles in several ways (20, 21) [for exam-ple, by imposing a curvature to the space (22) or failing to unifydifferent scales (23)]. The way that we conceive space, however,is not necessarily constrained by our perception: following Kant’sproposal, the axioms of Euclidean geometry may constitute themost intuitive conceptualization of space not only in adults ed-ucated in the tradition of Euclidean geometry (24) but also incultures where this tradition is absent. In particular, humansuniversally may be able to understand the aspects of Euclideangeometry that transcend perceptual and motor experience [forexample, objects that are either infinitely small (a line with nowidth) or infinitely large (an unbounded space), or statementsabout necessary or impossible configurations].To address the issue of the universality of Euclidean concepts

empirically, we probed intuitions of geometry in participantsfrom an indigene group in the Amazon, the Mundurucu (5, 25,26), who had had no access to education in geometry. Besidesthe lack of direct instruction, the Mundurucu’s experience ofspace differs from that of people in industrialized cultures inmany ways: for example, the Mundurucu engage daily in navi-gation tasks far more challenging than people living in a urbanenvironment, whereas their language does not lexicalize essentialconcepts in Euclidean geometry such as right angles or paral-lelism (27). Our previous research showed that the Mundurucuare sensitive to a host of sophisticated geometrical properties invisual images, including angles, alignment, and parallelism (5).However, it did not test whether the Mundurucu can reasonabout idealized, nonperceptible spatial objects and whether theyentertain thoughts of necessity and possibility concerning thoseobjects. Here, we asked three questions. (i) Do the Mundurucuunderstand that some straight lines may never cross? (ii) Do theypossess an intuition that the sum of three angles in a triangle isa constant? (iii) Are they aware that these properties are trueonly for points and lines on the plane, and would they flexiblyadapt their answers if the same questions were asked of geo-metrical figures drawn on a curved surface such as a sphere?

ResultsProbing Intuitive Knowledge of Straight Lines. Twenty-two adultand eight child participants from three isolated villages in theMundurucu territory were introduced to two idealized worlds,shaped either as a plane or as a sphere, and then were askedquestions about the properties of straight lines on each of thesesurfaces. Each subtest started with a short description of thesurface considered, either as a flat, endless world for the plane or

Author contributions: V.I., P.P., E.S.S., and S.D. designed research; V.I. and P.P. performedresearch; V.I. and S.D. analyzed data; and V.I., P.P., E.S.S., and S.D. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1V.I. and P.P. contributed equally to this work.2To whom correspondence should be addressed. E-mail: veronique.izard@polytechnique.org.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1016686108/-/DCSupplemental.

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as a world that is very round for the sphere. To help visualization,participants were referred to a real planar or spherical object(a table or a half-calabash) and illustrations presented on a com-puter screen (Fig. S1). The experimenter further explained thatvillages (corresponding to points) were present at the surface ofthese worlds as well as straight paths allowing people to walkdirectly from one village to another without turning (corre-sponding to straight lines). Illustrations of a plane or sphere withpoints and line segments were presented on the computer screento support the narration. After this introduction, participantswere asked a series of questions that were illustrated by sketchespresented on the computer screen (Table 1). The responsespredicted by planar and spherical geometry were identical for 12of the questions (for example, “can more than two lines be drawnthrough a given point?”), whereas they differed for the remain-ing 9 questions (for example, “can a line be drawn througha point so as not to intersect another line?”).Overall, the Mundurucu performed well above chance level

[83.0% correct, t(29) = 35.5, P < 0.0001]. There was no differ-ence between the adults (82.3%) and children [85.0%; F(1,28) =1.1, P = 0.30], but in both groups, accuracy was higher for theplane (adults = 93.1%; children = 95.8%) than for the sphere[adults = 71.5%; children = 74.1%; F(1,28) = 161.6, P <0.0001]. We separated the questions that called for identical ordistinct answers in planar and spherical geometry and analyzedthe percentage of responses that followed the predictions of

planar geometry. The Mundurucu modulated their responsesaccording to the embedding geometry only where needed,resulting in an interaction between condition (plane or sphere)and question type [F(1,28) = 116.3, P < 0.0001] that did notinteract with age group (F < 1) (Fig. 1). Nevertheless, in both agegroups and on both surfaces, the planar response was chosenmost often (comparison of the percentage of plane responses vs.chance, ps < 0.001). The pattern of responses was not affected bythe level of instruction for adults or by age for children (SI Textand Figs. S2–S4).To evaluate further the effect of experience, we tested, on the

same task, two groups of 16 adults from the United States and 8children from France that were matched in age with the Mun-durucu children. Both groups showed the same profile of per-formance as the Mundurucu (overall bias to plane answers: ps <0.001; interaction between subtest and question type: ps < 0.01;no triple interaction of group, subtest, and question type in ageneral ANOVA including both Mundurucu age groups and thegroups from the United States and France: F < 1).It could be objected that the general bias to planar answers,

present in both populations, was induced by the sketches pre-sented, because these were always drawn on a plane (in the caseof the sphere, a zooming animation gave the impression ofcoming so close to the surface that it appeared flat). However,this consideration would imply that all aspects of spherical ge-ometry are equally counterintuitive, which was not the case. In

Table 1. Lines intuitions test

This table lists the questions in the order that they appeared in the test together with the sketches presented to the participants and thepercentage of positive (yes) responses in the Mundurucu population. The same sketches were presented in both planar and spherical subtests,except for the last question. Table S2 shows the response rates in all other groups.*Questions that call for different answers in planar and spherical geometry.†Questions about intersections occurring in both directions.‡Questions about parallelism.

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the spherical subtest, the participants relied sometimes on therules of planar geometry and sometimes on correct reasoning onthe sphere, which was revealed by a close examination of thequestions whose theoretical answer contradicted planar geome-try. Participants of both cultures incorrectly asserted that somelines never cross, reaffirming the existence of parallel lines (threequestions; Mundurucu adults and children = 91.9% planaranswers; US adults = 97.9%; French children = 93.8%). How-ever, they were more likely to overcome their planar intuitionswhen stating that, on the sphere, lines may cross in both direc-tions (five questions; Mundurucu adults and children = 42.4%planar answers; US adults = 20.0%; French children = 52.5%),which yielded a significant interaction between the condition andtype of critical question [F(1,50) = 115.0, P < 0.0001]. The maininteraction was further qualified by a triple interaction withgroup [F(3,50) = 2.9, P = 0.046], because the French childrendid not revise their planar intuitions as much as the other groups.

Probing Intuitive Knowledge of Triangles. In our first task, we foundthat, in the absence of formal education in geometry, Mundur-ucu children and adults are able to reason about ideal conceptsin accordance with the predictions of Euclidean geometry. Tocomplement this first finding, in a second task, we probed quan-titative predictions in geometry. On each trial, participants wereshown an incomplete triangle on a planar or spherical surfaceconsisting of the two points of the base and two adjacent anglesending in arrows pointing to the triangle’s unseen apex (Fig. 2A).They were asked to indicate the position of the missing apex onthe screen and then to estimate the angle formed by the twoincomplete lines at their intersection. In two subsequent versionsof the task, Mundurucu participants indicated their angle esti-mation by positioning their hands in an inverted V shape or bymeans of a custom-made goniometer placed on the table (Fig.2B). All of the data were pooled together after preliminaryanalyses did not reveal any essential differences between the twotypes of responses (SI Text and Fig. S5).In the four groups of Mundurucu adults, Mundurucu children,

US adults, and French children, the responses followed the lawsof planar and spherical geometry closely. The coordinates of thereported apex and the estimated apex angles correlated well withthe values predicted, respectively, by planar or spherical geom-etry in the corresponding subtests (average r2 ranged between0.37 and 0.96 depending on the measure, group, and subtest)(Table S1). To assess whether the angles reported conformedbetter with the planar or spherical equations in each subtest, theangles estimated by each individual were entered in a multipleregression analysis with two predictors for the planar andspherical equations. An analysis of variance performed over theregression weights yielded a significant interaction betweencondition and predictor type [F(1,53) = 148.7, P < 0.0001], in-

dicating that children’s and adults’ angular responses in bothplane and sphere worlds were best predicted by the equations ofthe corresponding geometries (Fig. S6). This effect was presentin all of the groups tested (ps < 0.05) but nonetheless, interactedwith the group factor [F(3,53) = 3.7, P = 0.018], because bothMundurucu groups modulated their responses to a greater extentthan the US and French controls [adults: F(1,39) = 6.8, P =0.013; children: F(1,14) = 4.9, P = 0.045]. The bias to planarresponses observed in the US and French participants might berelated to their extensive familiarity with planar geometry or theextra care taken to familiarize Mundurucu participants with theshapes and task.To illustrate these findings, we computed the sum of the

reported angle and the two given angles (Fig. 2C). In theory, thissum is constant in planar geometry (180° or π radians), whereasin spherical geometry, it is greater than 180° and varies in apredictable way with the area of the triangle. The measured sumof the angles of the triangle conformed to these predictions: inthe planar subtest, the measured sum was essentially invariantand remarkably close to 180° [average sum between 179.9° and184.1° across groups; t tests vs. 180° ps > 0.5 except for the groupof French children: t(7) = 2.7, P = 0.031], whereas it was largerthan 180° in the spherical subtest (average sum between 189.3°and 201.5° across groups; ps < 0.05).

Geometrical Intuitions in US Children. The geometrical intuitionsuncovered by our tests could either belong to the core knowledgethat we inherit from evolution or be learned through interactionswith the environment (2). To probe the development of thisknowledge, we tested younger children aged 5 and 6 y in theUnited States. In the lines intuitions test, children’s responseswere significantly biased to planar responses (73.1% and 67.7%of planar responses, respectively, in the planar and sphericalsubtests; P < 0.004), but this bias was markedly less strong thanthe bias shown by the other groups [F(4,81) = 11.6, P < 0.0001].Moreover, the younger children did not adapt their responses tothe different surfaces or types of questions [no effect of subtestor type of question: Fs < 1; no interaction between type ofquestion and subtest: F(1,30) = 1.4, P = 0.25] (Fig. 3 Left).On the triangle test, children placed the intersection point

rather accurately on the plane and sphere (average individual r2between 0.28 and 0.74; ps < 0.05) (Table S1), but their capacityto generate the appropriate angle was limited (average r2 = 0.47and 0.18 on the plane and sphere, respectively; ps = 0.45 and0.021, respectively). To better understand the results of thecorrelation and the strategies used by children, we used a generallinear model including regressors for the distance between thetwo given points, the sum of the base angles, and the type ofsurface (planar vs. spherical subtest)—all factors that shouldtheoretically impact the responses. Although, as suggested by the

Fig. 1. Performance in the lines intuitions test. Percentage of planar responses to questions that call for the same answer or distinct answers in the plane andsphere subtests (labels same or dis on the x axis). Perfect mathematical understanding calls for 100% planar responses, except for the distinct answerquestions on the sphere, where there should be 0% planar responses (last bar; shown in white). The results partially conform to this pattern but with a strongbias to planar responses.

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previous analyses, all of the effects predicted by the equations ofEuclidean geometry were observed in the Mundurucu partic-ipants, the US adults, and to a lesser extent, the French children(SI Text), young US children’s angle estimates depended exclu-sively on the distance between the two base points [F(1,16) =29.8, P < 0.0001] without taking into account the two anglesgiven at the base or the type of surface (Fs < 1) (Fig. 3 Right).Because the distance between the base points influences the apexangle only in the equations of spherical geometry, the observedcorrelation between children’s estimates and correct predictionson the sphere seems to be a fortuitous consequence of children’sgeneral heuristics based on distance, and it does not indicate abetter understanding of angles on the sphere than on the plane.Together, these results show that children have an early bias to

Euclidean intuitions of space, but this initial bias does not seemto encompass all principles of Euclidean geometry.

DiscussionAfter a fewminutes of pedagogical introduction describing points,lines, and surfaces as metaphorical villages, routes, and land-scapes, we elicited remarkably accurate and flexible Euclideanintuitions in people who had received no formal instruction ingeometry. In two tests, participants combined spatial information

given verbally and visually and generated appropriate normativeand quantitative geometric predictions. Together, the results ofboth tests suggest that Euclidean geometry, inasmuch as it con-cerns basic objects such as points and lines on the plane, is a cross-cultural universal that results from inherent properties of thehuman mind as it develops in its natural environment.In the first test, the Mundurucu were asked to answer a series

of questions about possible and impossible configurations oflines and points on a flat or spherical surface. Crucially, some ofthe questions required the participants to reason about conceptsthat outrun the limits of sensorimotor experience, such asinfinite, parallel lines and statements of impossibility. Mundur-ucu adults and children achieved near-perfect performance onthe plane and to some extent, also adapted their responses to fitthe properties of the spherical surface. The responses of theparticipants showed that they conceptualized the plane as infi-nite, such as when they stated that it is possible to draw a thirdparallel to two parallel lines (87.9% yes responses) but is impos-sible to draw a line that crosses only one of two parallels (12.1%yes responses).In a second test about quantitative predictions, participants

were requested to estimate the position and angle of the missingpart of an incomplete triangle. To be accurate with respect to the

Fig. 2. The triangle completion test. (A) Stimuli and (B)response mode in 2006 (hands) and 2007 (goniometer).(C) Sum of the angles of the triangles estimated by theparticipants. Each dot represents a different problemindexed along the x axis according to the value of thepredicted sum in spherical geometry (in planar geometry,the predicted sum is constant and equal to 180°). Thedotted lines correspond to the values predicted by planarand spherical geometry, and the plain lines correspond tolinear fits of the data. (Photos copyright Pierre Pica &CNRS.)

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equations of Euclidean geometry, they needed to take into ac-count a variety of parameters: the base length of the triangle andthe base angles as well as the curvature of the surface. Thissecond task also required a good understanding of straight linesto generate valid extrapolation of the sides of the triangle. Onthe planar surface, Mundurucu adults and children producedangles such that the summed angles of the triangle fell close to180°, in accordance with a central tenet of Euclidean geometry(the constancy of the sum of angles in any planar triangle). In-deed, it is rather remarkable that approximately two decimals ofπ could be obtained by merely asking Amazonians to estimatethe angles in a triangle. Furthermore, when they were given thesame task on a spherical surface, our Mundurucu participantsmodulated the size of the created angle with the area of thetriangle in accord with the equations of spherical geometry, thusproducing larger angles than in the planar case.Despite wide differences in the type of spatial experience avail-

able in these cultures, the performance of our Mundurucu par-ticipants converged remarkably with that of geometry-educated,urban control groups from the United States and France. In linewith our previous research on intuitive arithmetic and geometry inthe Mundurucu (5, 25, 26) and with Plato’s views on education asdeveloped in the Meno, the present results indicate that sophisti-cated protomathematical intuitions for both arithmetic and ge-ometry can be revealed in all humans provided that the relevantabstract concepts are exemplified by concrete situations. Ourimaginary worlds of villages and paths played this role, sup-porting the communication between the experimenter and theparticipants and perhaps, also contributing causally to awakenabstract concepts of geometry.In our previous research on numeric cognition, we observed

a mosaic of convergences and divergences between the Mun-durucu and US or French populations (25, 26, 28). Divergenceswere observed in tasks that typically elicit the use of culture-specific tools such as counting or linear measurement. Similarly,in the present tests, the US adults and French children showeda stronger bias to plane geometry, which may reflect culture-specific knowledge of planar Euclidean geometry. Interestingly,Mundurucu participants and young US children (tested beforethe onset of education in geometry) were not devoid of a planarbias, although in both cases, this bias was restricted to the nor-mative reasoning task. The human mind may be intrinsicallymore prepared to reason about flat surfaces, perhaps because flatsurfaces are computationally simpler, as originally suggested byPoincaré, or because they are most frequently encountered in

navigable environments. In either case, this effect is exaggeratedin populations educated to Euclidean planar geometry.Despite the cross-cultural convergence in adults and school-

aged children, geometrical intuitions do not seem to be fully inplace by 5 or 6 y of age. Tested on the same tasks, young USchildren showed some abstract intuitions of Euclidean geometry,particularly when reasoning about the properties of straight lineson a plane; however, they were less able to adapt their responsesto different surfaces, and they failed to produce angles thatrespected the constancy of the sum of three angles in a triangle.This last finding echoes recent reports of failure to use angles intoddlers and young preschool children (29, 30), which came asa surprise given the success of children in navigational trianglecompletion tasks (15) and their ability to process angles in 2Dsmall-scale figures from infancy (31–33). Although more re-search is needed to explore why the young children sometimesfail with angles, one possibility is that the adult concept of ab-stract angle, applying equally well to planar figures and 3Dshapes and to small- and large-scale figures, is a mental constructarising during childhood (34). By 5 or 6 y, this construct may stillbe too fragile to be applied in tasks such as those in the presentstudy, where the crucial information is given across differentformats, partly in a verbal description that takes a navigationalperspective (as in paths that always go straight ahead) and partlyin bird’s eye views of surfaces.By adulthood, however, concepts of angle and parallelism

obviously become entrenched even in the absence of formaleducation. Future research should evaluate the role of spatialexperience in this emergence (15, 35). Abstract geometry may beinnate but emerge only after a certain point in development, or itmay be learned on the basis of a type of experience with spacethat is so general that it is encountered by all human beings. Wehave no evidence bearing on the choice between these twopossibilities, but the hypothesis that experience triggers or shapesour Euclidean intuitions is worth exploring. Such explorationsmay address a general puzzle that extends back to Plato: how canexperience with the concrete world of perception and actionyield a set of concepts that radically outruns what we can per-ceive or do?

Materials and MethodsParticipants. The present data come from a total of 30 Mundurucu partic-ipants tested during twofield trips undertaken by P.P. in 2006 and 2007 (threeof the participants were tested during both visits). All of the Mundurucuparticipants came from an isolated area of the Mundurucu Territory (up to150 km upstream of the Cururu mission; map shown in Fig. S7). They were allnative Mundurucu speakers, and none had received formal instruction ingeometry; 8 participants were children (ages 7–13 y, average = 10.0 y, fivemales), and 22 were adults (ages 15–75 y, average = 37.9 y, eight males).

All of theMundurucu participants completed both the triangle completiontest and the lines intuitions test. The lines intuitions test was always presentedfirst to serve as an introduction to the geometrical properties of the shapes.The order of presentation of the planar and spherical subtests was ran-domized for the triangle completion task. In the lines intuitions task, mostparticipants were tested on the plane subtest first.

Two different groups of adult participants from the United States (Group1: n = 19, ages 19–58 y, average = 33.4 y, eight males; Group 2: n = 16, ages16–58 y, average = 37.9 y, nine males) served as controls, respectively, in thetriangle and lines intuitions tasks. One group of eight children participantsfrom France (ages 7–13 y, average = 10.25 y, four males, attending firstthrough seventh grade), matched in age with the Mundurucu children, wastested on both tasks. Just like the Mundurucu, the children were tested onthe lines intuitions task first. The order of presentation of the planar andspherical subtests was randomized in both tasks for the control groups.

Finally, we tested two groups of United States children (Group 1: n = 20,ages 5.0–6.9 y, average = 6.0 y, 8 males; Group 2: n = 32, ages 5.1–7.0 y,average = 5.9 y, 15 males), respectively, in the triangle completion task andthe lines intuitions task. Children attended preschool, kindergarten, or firstgrade in the greater Boston area (this information is not available for threeof the children). Two additional children were excluded from the finalsample in the lines intuitions task, one because he was attending secondgrade despite his young age and the other because he refused to use themouse of the computer and failed to give clear answers. Each child partici-

Fig. 3. Performance of young US children. (Left) Percentage of planarresponses in the lines intuitions test as in Fig. 1. (Right) Estimated sum ofangles in the triangle completion test as in Fig. 2. Error bars represent SE.

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pated only in either the planar or spherical subpart in the triangle com-pletion or lines intuitions tests.

Lines Intuitions Test. At the beginning of each subtest, participants wereintroduced to an ideal world, described either as a very flat, never-endingworld (plane) or a very round world like a ball (sphere). Power point slidespresented on a computer served as an illustration (Fig. S1). Participants werealso referred to a real flat or round-shaped object. The text introduced theexistence of villages (represented as dots in the power point slides) andpaths (represented as lines). To convey the constraints that all lines werestraight, the participants were told that the paths were always going verystraight, always in front of them; they were told also that paths neverended. A computer animation, programmed in Visual Python, showed anexample of a path traced on a plane or a sphere. Finally, the slides illustratedthe possibility for paths to go through villages.

After this introduction, participants were asked a series of 21 ques-tions pertaining to the properties of lines and dots (Table 1). The samequestions were used in both planar and spherical subtests in a fixed order.Questions were illustrated by sketches of lines and dots presented ona computer. Crucially, these sketches were strictly identical in both subtests:a zoom effect created the illusion of coming so close to the surface that eventhe sphere appeared flat. The questions were read aloud to the participantsby a trained Mundurucu, English, or French speaker. The translation of thetext in Mundurucu is available in SI Text. For the US children, the script wasadapted to encourage answers: questions were presented as constructionprojects, proposed by a potentially silly king, to be judged feasible or non-feasible. US children gave their response by clicking on one of two mockresponse buttons representing a thumb up or down; other participants gavetheir responses orally.

Triangle Completion Test. In each triangle completion problem, two villageswere shown on a plane or at the base of a half-sphere, with arrows pointingin the direction of a third village (unseen) (Fig. 2A). In the spherical subtest,the sphere rotated to present both villages successively in the center ofthe screen, allowing a better estimation of the shape of the surface and theangle spanned. Participants were required to estimate the location of thisthird village by pointing on the screen and then, to estimate the angleformed by the two interpolated paths at the intersection. To do so, in 2006,the Mundurucu held their two hands in an inverted V shape to reproduce

the estimated angle, and the experimenter (P.P.) measured the aperturewith a goniometer. In the 2007 replication study as well as in the US andFrench studies, participants reported the angle themselves using a custom-made goniometer placed horizontally on the table in front of the computer.The results from these two versions were pooled after analyses revealed nosubstantive differences in the results (SI Text and Fig. S5). Before testing,participants were asked to reproduce three to six angles from a card to makesure that they were able to use the goniometer.

Participants were first given two training trials with feedback. YoungerUS children were also given two additional training trials with feedback ona real plane or half-sphere object using dots and arrows mounted on Velcro.After this training phase, 25 test trials presented in a random order spanneda variety of triangular configurations (without feedback). The same trialparameters were used in both subtests. Mundurucu participants performedbetween 16 and 93 trials in each subtest (on average, 33.3 and 31.3 trials inthe planar and spherical subtests, respectively); because of technical problemswith the heating of the computer, the experiments were occasionally restartedfor some participants, resulting in a larger number of trials. For the purpose ofthe analyses, datawere averagedwithin participants for each trial type. The USadult and French child control participants each contributed 25 trials of eachtype. Younger US children performed, respectively, 23.0 or 23.9 trials on av-erage in the planar and spherical tests.

ACKNOWLEDGMENTS. This work is part of a larger project on the nature ofquantification. It is based on psychological experiments and linguisticsstudies conducted in the Mundurucu territory (Pará, Brazil) under thedirection of P.P. in accordance with the Consehlo de Desenvolvimento Cien-tifico et Tecnologicico and the Fundação do Indio (Funaï; Processo 2857/04).We thank L. Braga (SARAH Network of Neurorehabilitation Hospitals,Brazil), A. Ramos (Coordenação Geral de Educação, Funaï), and C. Romeiro(Nucleo de Documentação e Pesquisa, Funaï) for discussion and advice;M. Karu, Y.-H. Liu, and C. Tawe for help with data collection; P. Stephenson(The Array Rainforest Foundation) for the satellite map of the Mundurucuterritory; and D. Sutherland and T. R. Virgil for comments on the manuscript.This work was supported by the Institut National de la Santé et de laRecherche Médicale, the Fyssen Foundation (to V.I.), the Département desSciences Humaines et Sociales of Centre National de la Recherche Scientifi-que (to P.P.), the National Institutes of Health (to E.S.), a McDonnell Foun-dation centennial fellowship (to S.D.), and a Starting Grant from theEuropean Research Council (MathConstruction 263179) (to V.I.).

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